Topology of Stratified Spaces MSRI Publications Volume 58, 2011 Characteristic classes of mixed Hodge modules JORG¨ SCHURMANN¨ ABSTRACT. This paper is an extended version of an expository talk given at the workshop “Topology of Stratified Spaces” at MSRI in September 2008. It gives an introduction and overview about recent developments on the in- teraction of the theories of characteristic classes and mixed Hodge theory for singular spaces in the complex algebraic context. It uses M. Saito’s deep theory of mixed Hodge modules as a black box, thinking about them as “constructible or perverse sheaves of Hodge struc- tures”, having the same functorial calculus of Grothendieck functors. For the “constant Hodge sheaf”, one gets the “motivic characteristic classes” of Brasselet, Schurmann,¨ and Yokura, whereas the classes of the “intersection homology Hodge sheaf” were studied by Cappell, Maxim, and Shaneson. The classes associated to “good” variation of mixed Hodge structures where studied in connection with understanding the monodromy action by these three authors together with Libgober, and also by the author. There are two versions of these characteristic classes. The K-theoretical classes capture information about the graded pieces of the filtered de Rham complex of the filtered D-module underlying a mixed Hodge module. Appli- cation of a suitable Todd class transformation then gives classes in homology. These classes are functorial for proper pushdown and exterior products, to- gether with some other properties one would expect for a satisfactory theory of characteristic classes for singular spaces. For “good” variation of mixed Hodge structures they have an explicit clas- sical description in terms of “logarithmic de Rham complexes”. On a point space they correspond to a specialization of the Hodge polynomial of a mixed Hodge structure, which one gets by forgetting the weight filtration. We also indicate some relations with other subjects of the conference, like index theorems, signature, L-classes, elliptic genera and motivic characteristic classes for singular spaces. 419 420 JORG¨ SCHURMANN¨ CONTENTS 1. Introduction 420 2. Hodge structures and genera 423 3. Characteristic classes of variations of mixed Hodge structures 430 4.CalculusofmixedHodgemodules 440 5.CharacteristicclassesofmixedHodgemodules 450 Acknowledgements 467 References 467 1. Introduction This paper gives an introduction and overview about recent developments on the interaction of the theories of characteristic classes and mixed Hodge theory for singular spaces in the complex algebraic context. The reader is not assumed to have a background on any of these subjects, and the paper can also be used as a bridge for communication between researchers on one of these subjects. General references for the theory of characteristic classes of singular spaces are the survey [48] and the paper [55] in these proceedings. As references for mixed Hodge theory one can use [38; 52], as well as the nice paper [37] for explaining the motivic viewpoint to mixed Hodge theory. Finally as an intro- duction to M. Saito’s deep theory of mixed Hodge modules one can use [38, Chapter 14], [41] as well as the introduction [45]. The theory of mixed Hodge modules is used here more or less as a black box; we think about them as constructible or perverse sheaves of Hodge structures, having the same functorial calculus of Grothendieck functors. The underlying theory of constructible and perverse sheaves can be found in [7; 30; 47]. H For the “constant Hodge sheaf” QZ one gets the “motivic characteristic classes” of Brasselet, Schurmann,¨ and Yokura [9], as explained in these pro- H ceedings [55]. The classes of the “intersection homology Hodge sheaf” ICZ were studied by Cappell, Maxim, and Shaneson in [10; 11]. Also, the classes associated to “good” variation of mixed Hodge structures where studied via Atiyah–Meyer type formulae by Cappell, Libgober, Maxim, and Shaneson in [12; 13]. For a summary compare also with [35]. There are two versions of these characteristic classes, the motivic Chern class transformation MHCy and the motivic Hirzebruch class transformation MHTy. The K-theoretical classes MHCy capture information about the graded pieces of the filtered de Rham complex of the filtered D-module underlying a mixed Hodge module. Application of a suitable twisting td.1Cy/ of the Todd class transformation td of Baum, Fulton, and MacPherson [5; 22] then gives the classes MHTy D td.1Cy/ıMHCy in homology. It is the motivic Hirzebruch class transformation MHTy, which unifies three concepts: CHARACTERISTICCLASSESOFMIXEDHODGEMODULES 421 .y D 1/ the (rationalized) Chern class transformation c of MacPherson [34]; .y D 0/ the Todd class transformation td of Baum–Fulton–MacPherson [5]; .y D 1/ the L-class transformation L of Cappell and Shaneson [14]. (Compare with [9; 48] and also with [55] in these proceedings.) But in this paper we focus on the K-theoretical classes MHCy, because these imply then also the corresponding results for MHTy just by application of the (twisted) Todd class transformation. So the motivic Chern class transformation MHCy studied here is really the basic one! Here we explain the functorial calculus of these classes, first stating in a very precise form the key results used from Saito’s theory of mixed Hodge modules, and then explaining how to get from this the basic results about the motivic Chern class transformation MHCy. These results are illustrated by many inter- esting examples. For the convenience of the reader, the most general results are only stated near the end of the paper. In fact, while most of the paper is a detailed survey of the K-theoretical version of the theory as developed in [9; 12; 13; 35], it is this last section that contains new results on the important functorial properties of these characteristic classes. The first two sections do not use mixed Hodge modules and are formulated in the now classical language of (variation of) mixed Hodge structures. Here is the plan of the paper: SECTION 2 introduces pure and mixed Hodge structures and the corresponding Hodge genera, such as the E-polynomial and the y-genus. These are suit- able generating functions of Hodge numbers with y using only the Hodge filtration F, whereas the E-polynomial also uses the weight filtration. We also carefully explain why only the y-genus can be further generalized to characteristic classes, i.e., why one has to forget the weight filtration for ap- plications to characteristic classes. SECTION 3 motivates and explains the notion of a variation (or family) of pure and mixed Hodge structures over a smooth (or maybe singular) base. Basic examples come from the cohomology of the fibers of a family of complex al- gebraic varieties. We also introduce the notion of a “good” variation of mixed Hodge structures on a complex algebraic manifold M , to shorten the notion for a graded polarizable variation of mixed Hodge structures on M that is ad- missible in the sense of Steenbrink and Zucker [50] and Kashiwara [28], with quasi-unipotent monodromy at infinity, i.e., with respect to a compactification M of M by a compact complex algebraic manifold M , with complement D WD M nM a normal crossing divisor with smooth irreducible components. Later on these will give the basic example of so-called “smooth” mixed Hodge modules. And for these good variations we introduce a simple cohomological characteristic class transformation MHCy, which behaves nicely with respect 422 JORG¨ SCHURMANN¨ to smooth pullback, duality and (exterior) products. As a first approximation to more general mixed Hodge modules and their characteristic classes, we also study in detail functorial properties of the canonical Deligne extension across a normal crossing divisor D at infinity (as above), leading to cohomo- y logical characteristic classes MHC .j. // defined in terms of “logarithmic de Rham complexes”. These classes of good variations have been studied in detail in [12; 13; 35], and most results described here are new functorial reformulations of the results from these sources. SECTION 4 starts with an introduction to Saito’s functorial theory of algebraic mixed Hodge modules, explaining its power in many examples, including how to get a pure Hodge structure on the global intersection cohomology IH .Z/ of a compact complex algebraic variety Z. From this we deduce the basic calculus of Grothendieck groups K0.MHM. // of mixed Hodge modules needed for our motivic Chern class transformation MHCy. We also explain the relation to the motivic viewpoint coming from relative Grothen- dieck groups of complex algebraic varieties. SECTION 5.1 is devoted to the definition of our motivic characteristic homology class transformations MHCy and MHTy for mixed Hodge modules. By Saito’s theory they commute with push down for proper morphisms, and on a compact space one gets back the corresponding y-genus by pushing down to a point, i.e., by taking the degree of these characteristic homology classes. SECTIONS 5.2 AND 5.3 finally explain other important functoriality properties: (1) multiplicativity for exterior products; (2) the behavior under smooth pullback given by a Verdier Riemann–Roch formula; (3) a “going up and down” formula for proper smooth morphisms; y (4) multiplicativity between MHC and MHCy for a suitable (co)homological pairing in the context of a morphism with smooth target (as special cases one gets interesting Atiyah and Atiyah–Meyer type formulae, as studied in [12; 13; 35]); (5) the relation between MHCy and duality, i.e.,